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StyleBox["Lab 01 - Graphing Functions of Two Variables and Quadric Surfaces",
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"\nMath 2374 - University of Minnesota\nhttp://www.math.umn.edu/math2374\n\
Questions to: rogness@math.umn.edu"
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"At the end of Lab 1A you learned how to use ",
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" to plot the graph of equations such as ",
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FormBox[
RowBox[{
RowBox[{
SuperscriptBox["x", "2"], "+",
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". As you hopefully remember, we have to use different commands to plot the \
graphs of these two equations. In the first case, we have ",
StyleBox["explicitly",
FontSlant->"Italic"],
" solved for y as a function of x; there is a single y on the left hand side \
of the equation, and no occurrences of y on the right hand side. In cases \
like this we can use the ",
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" function to show a graph of y. In the second case we have an ",
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" function of y. We can't solve explicitly for y because we end up with \
\[PlusMinus]",
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"on the right hand side. (This is not a well defined function because for a \
given value of x we can only have one value, not a positive ",
StyleBox["and",
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" a negative value.) We learned how to use the command ",
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FontWeight->"Bold"],
" to handle equations like this.\n\nIn this lab we're going to work with the \
three-dimensional analogs of these commands and, as you might expect, we'll \
have to consider two different cases. The first is when we have a function \
of x and y which is explicitly solved for z, e.g.\n\n",
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"\n\nWe'll also consider implicit functions of z, such as\n\n",
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"\n\n(Note that if you tried to solve this last equation for z, you'd have \
the same problem with a \[PlusMinus] sign.)\n\nThe first case is considerably \
easier, and we'll deal with that one first. One other comment before we move \
on: some of the equations in this lab might look a bit small on your screen. \
If you're having trouble reading them, try the menu option Format : Screen \
Environment : Presentation."
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Cell["Suppose we define a function z=f(x,y) of two variables, e.g.", "Text"],
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"What does a graph of this function even ",
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"? The definition usually given is \"the set of all points of the form ",
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" such that ",
StyleBox["(x,y)",
FontSlant->"Italic"],
" is in the domain of ",
StyleBox["f",
FontSlant->"Italic"],
",\" but you may not find this particularly enlightening.\n\nNotice that the \
function f takes two inputs, x and y, and returns a single number, which we \
call z. If we draw the x-y-z coordinate axes in the standard way, the z-axis \
represents height, and this is the key to graphing ",
StyleBox["f(x,y)",
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". If you choose a point ",
StyleBox["(x,y)",
FontSlant->"Italic"],
" in the xy-plane, then ",
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FontSlant->"Italic"],
" represents the height of the graph at that point. For example, here's the \
graph of a simple function, g(x,y)=1. This means that no matter what values \
you choose for x and y, the function g will always return (\"a height of\") \
one."
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"As you can see, we're using the command ",
StyleBox["Plot3D",
FontWeight->"Bold"],
" to create this graph. The syntax of ",
StyleBox["Plot3D",
FontWeight->"Bold"],
" is very similar to that of ",
StyleBox["Plot",
FontWeight->"Bold"],
"; you first give it a function of x and y, and then ranges for x and y."
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StyleBox["3D Graph Controls",
FontSize->16,
FontWeight->"Bold"],
"\n\nIn version 6.0, ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" finally implemented interactive controls for its 3D graphics. You can \
click and drag the picture above to rotate it. If you hold the control key \
while clicking on the picture and dragging up/down you will zoom in/out of \
the picture. Holding the shift key while dragging will move the picture \
around the window. (Note that the ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" documentation says to use shift to zoom and control to move the picture, \
so if you have problems try both keys.)"
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Cell["\<\
Here's a slightly more complicated function. Before you evaluate this cell, \
see if you can predict what the graph will look like.\
\>", "Text",
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Cell[TextData[{
"This graph does not have a constant height of one, but if you look at the \
definition of g(x,y) you should be able to make some observations:\n\n\
\[FilledVerySmallSquare] if ",
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", then g(x,y)=0.\n\[FilledVerySmallSquare] if x and y are both positive, \
then ",
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Cell[BoxData[
FormBox[
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".\n\nLooking back at the picture, are these things true? One thing you \
have to notice is that the z-axis goes from -2 to 2, which means the \"height\
\" is zero halfway up the box, not at the bottom. Another thing worth \
pointing out is that the z-axis is scaled differently than the x- and y-axes. \
If you want to change the scaling, you can use the option ",
StyleBox["BoxRatios",
FontWeight->"Bold"],
", which you can look up in the Help Browser",
":"
}], "Text"],
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Cell[TextData[{
"Let's plot the function ",
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" we considered previously so that we have something slightly more \
interesting to work with. "
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Cell[TextData[{
"Another useful option is ",
StyleBox["ViewPoint",
FontWeight->"Bold"],
", which allows you to specify the location of your \"eyes\" as you look at \
the surface. For example, the following command puts you at the point \
(0,0,10) on the ",
StyleBox["z",
FontSlant->"Italic"],
"-axis, looking down at the surface."
}], "Text",
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Cell[TextData[{
StyleBox["ViewPoint",
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" isn't so necessary now that you can rotate 3D graphics in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" -- you can just plot the picture and rotate it as needed. But you might \
find it helpful at times to specify the viewpoint so you don't always have to \
re-rotate the picture. \n\nWe could spend an entire lab having you plot the \
graphs of all sorts of functions. Some of the most interesting involve \
trigonometric functions like ",
StyleBox["Sin",
FontWeight->"Bold"],
":"
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Cell[TextData[{
"If you don't like the gridlines, you can turn them off with the ",
StyleBox["Mesh",
FontWeight->"Bold"],
" option:"
}], "Text"],
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Cell[TextData[{
"Sometimes when you're changing viewpoints it's easy to lose track of which \
axis is the x-axis, and which is the y-axis. You can label them with the ",
StyleBox["AxesLabel",
FontWeight->"Bold"],
" option:"
}], "Text"],
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"Or, if you don't want to have the axes numbered, you can use the ",
StyleBox["Axes",
FontWeight->"Bold"],
" option. (If you want the box to disappear entirely, try adding ",
StyleBox["Boxed\[Rule]False",
FontWeight->"Bold"],
" to this command.)"
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Cell[TextData[{
StyleBox["Exercise 1",
FontSize->16,
FontWeight->"Bold"],
"\n\nDownload the \"Addendum to Exercise 1\" for this lab from the course \
website. Use ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" to graph the functions in that document with the pictures below. You will \
make your life easier if you match the x- and y- ranges in the ",
StyleBox["Plot3D",
FontWeight->"Bold"],
" commands to the ranges in the figures. The ",
StyleBox["PlotPoints",
FontWeight->"Bold"],
" and ",
StyleBox["BoxRatios",
FontWeight->"Bold"],
" options might also be useful. Also note that the viewpoints in the \
figures might be different than the default viewpoint in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". \n\nIf you name the functions before plotting them, remember to use ",
StyleBox["lower-case",
FontSlant->"Italic"],
" names as discussed last week or you will run into problems.\n\nWhile we'd \
like to you to put some effort into identifying each graph, you will only \
hand in a careful write up one particular match. For example, if you're told \
to write about B(x,y), you should carefully explain why B(x,y) produces the \
graph that it does. Use the definition of the function to explain the shape \
-- why is it high in some areas, low in others? Is it ever equal to zero? \
Is it ever negative, or is it always positive? etc.\n\nYour TA will tell you \
which function has been chosen for you to describe."
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Cell[TextData[StyleBox["ContourPlot",
FontSize->16]], "Section"],
Cell[TextData[{
"Before we work with implicit functions, we're going to introduce one more \
way to examine the height of a surface. If you've ever done any kind of \
hiking outdoors or fishing on a big lake you're probably familiar with \
topographic maps. These maps use so-called \"contour lines\" to represent \
elevation. For instance, on standard United States Geological Survey maps, \
each contour line represents 10 feet of elevation. If you'd like to see an \
example of a topographic map, copy the following link and paste it into a web \
browser to see to a topographic map of Eagle Mountain, the highest mountain \
(well... hill) in Minnesota.\n\n\
http://www.dnr.state.mn.us/maps/tomo.html?mode=recenter&size=3&layer=24k&col=\
513&row=243\n\nThe contour lines on this map represent elevation above sea \
level. Notice how in some places the lines are very close together, which \
represents a steep slope. In other places the lines are further apart, which \
represents a more gradual slope. Not surprisingly, the steepest slopes seem \
to be very near the summits of Eagle Mountain and Moose Mountain.\n\n",
StyleBox["Mathematica",
FontSlant->"Italic"],
" can draw a topographic map of a surface for us. For example, let's look \
at a map of this function:"
}], "Text"],
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RowBox[{"y", "^", "2"}]}]}], ";"}]], "Input"],
Cell[TextData[{
"To draw the topographic map, or \"contour diagram,\" of the function, we \
use the command ",
StyleBox["ContourPlot",
FontWeight->"Bold"],
". We have to give the command the function we want to plot, and ranges for \
x and y:"
}], "Text"],
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"Notice that ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" shades the picture according to elevation. The darker regions represent \
the lower points; the lighter shades represent higher points. Scroll back \
and look at the graph of f(x,y) generated by ",
StyleBox["Plot3D",
FontWeight->"Bold"],
" and see if this contour diagram makes sense to you. Notice in particular \
that the contour lines get closer to each other near the edges of the \
diagram. Why is this and what does this mean? (If you're not sure, talk to \
the students next to you and/or your TA before you go on.)\n\nIf you have \
trouble remembering which points are high and which points are lower, move \
your mouse over the contour lines in the picture. ",
StyleBox["Mathematica",
FontSlant->"Italic"],
" will show you the \"elevation\" on each line. If you'd like to have these \
displayed at all times, use the ",
StyleBox["ContourLabels",
FontWeight->"Bold"],
" option:"
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Cell[TextData[{
"The term ",
StyleBox["ContourPlot",
FontWeight->"Bold"],
" is used in many different contexts in ",
StyleBox["Mathematica",
FontSlant->"Italic"],
". In the previous section we used it to graph the contour lines of a \
function, ",
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RowBox[{
RowBox[{"f", "(", "x", ")"}], "=",
RowBox[{
SuperscriptBox["x", "2"], "+",
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". Last week you used ",
StyleBox["ContourPlot",
FontWeight->"Bold"],
" to graph an implicit function, ",
Cell[BoxData[
FormBox[
RowBox[{
RowBox[{
SuperscriptBox["x", "2"], "+",
SuperscriptBox["y", "2"]}], "=", "1"}], TraditionalForm]]],
". There is a 3D analog called ",
StyleBox["ContourPlot3D",
FontWeight->"Bold"],
" which you can use to plot implicit functions with three variables."
}], "Text",
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Cell[TextData[{
"Suppose we want to graph an implicit function of z like this:\n\n",
Cell[BoxData[
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RowBox[{
RowBox[{
SuperscriptBox["x", "2"], "+",
SuperscriptBox["y", "2"], "+",
SuperscriptBox["z", "2"]}], "=", "1."}], TraditionalForm]]],
"\n\nWe use the following command, which is very similar to how we graphed \
",
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SuperscriptBox["x", "2"], "+",
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" with ",
StyleBox["ContourPlot",
FontWeight->"Bold"],
" last week; the only differences are the ",
StyleBox["3D",
FontWeight->"Bold"],
" at the end of the command together with a range for ",
StyleBox["z",
FontSlant->"Italic"],
":"
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RowBox[{"x", "^", "2"}], "+",
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RowBox[{"-", "1"}], ",", "1"}], "}"}], ",",
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Cell[TextData[{
"Notice that we chose the ranges for x, y, and z to go from -1 to 1. These \
ranges have a great influence over the resulting picture. If you make them \
too large, the command will run much faster but the picture will look awful. \
If you make them too small, you won't see your sphere (because you'll \
actually be ",
StyleBox["inside",
FontSlant->"Italic"],
" it). Try changing all of the ranges above to {_,-3,3} and {_,-1/2,1/2} to \
see examples of this.\n\nThe point is this: when you use ",
StyleBox["ContourPlot3D",
FontWeight->"Bold"],
" you should put careful thought into your ranges.\n\nOne other note: you \
can plot any kind of equation with x, y, and z using ",
StyleBox["ContourPlot3D",
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If you don't remember how to graph ellipses and hyperbolas, this might be a \
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Rather than have you read through a long section where we work out the cross \
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http://www.math.umn.edu/~rogness/quadrics/
The gallery also includes another short review about cross sections, along \
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following picture. You don't need to completely match every detail, but your \
answer should have contour lines which are not evenly spaced, and they should \
be ellipses, not circles or squares. Pay careful attention to the colors; \
recall that the shades from ",
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1 .5 L
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1 .74038 L
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1 .06731 L
s
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1 .11538 L
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1 .16346 L
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s
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1 1 L
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1 1 L
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closepath
clip
newpath
.3 g
.01923 .98077 m
.98077 .98077 L
.98077 .01923 L
.01923 .01923 L
F
0 g
.5 Mabswid
.2 g
.01923 .27652 m
.02336 .25962 L
.03444 .21955 L
.04755 .17949 L
.05929 .14822 L
.06286 .13942 L
.08066 .09936 L
.09936 .06278 L
.10127 .05929 L
.12512 .01923 L
.01923 .01923 L
F
0 g
.01923 .27652 m
.02336 .25962 L
.03444 .21955 L
.04755 .17949 L
.05929 .14822 L
.06286 .13942 L
.08066 .09936 L
.09936 .06278 L
.10127 .05929 L
.12512 .01923 L
s
.2 g
.01923 .72348 m
.02336 .74038 L
.03444 .78045 L
.04755 .82051 L
.05929 .85178 L
.06286 .86058 L
.08066 .90064 L
.09936 .93722 L
.10127 .94071 L
.12512 .98077 L
.01923 .98077 L
F
0 g
.01923 .72348 m
.02336 .74038 L
.03444 .78045 L
.04755 .82051 L
.05929 .85178 L
.06286 .86058 L
.08066 .90064 L
.09936 .93722 L
.10127 .94071 L
.12512 .98077 L
s
.1 g
.01923 .15925 m
.02654 .13942 L
.04293 .09936 L
.05929 .06418 L
.06173 .05929 L
.08331 .01923 L
.01923 .01923 L
F
0 g
.01923 .15925 m
.02654 .13942 L
.04293 .09936 L
.05929 .06418 L
.06173 .05929 L
.08331 .01923 L
s
.1 g
.01923 .84075 m
.02654 .86058 L
.04293 .90064 L
.05929 .93582 L
.06173 .94071 L
.08331 .98077 L
.01923 .98077 L
F
0 g
.01923 .84075 m
.02654 .86058 L
.04293 .90064 L
.05929 .93582 L
.06173 .94071 L
.08331 .98077 L
s
.01923 .07304 m
.02549 .05929 L
.04536 .01923 L
.01923 .01923 L
F
.01923 .07304 m
.02549 .05929 L
.04536 .01923 L
s
.01923 .92696 m
.02549 .94071 L
.04536 .98077 L
.01923 .98077 L
F
.01923 .92696 m
.02549 .94071 L
.04536 .98077 L
s
.4 g
.17223 .98077 m
.14518 .94071 L
.13942 .93124 L
.12217 .90064 L
.10252 .86058 L
.09936 .85352 L
.0857 .82051 L
.07142 .78045 L
.05943 .74038 L
.05929 .7399 L
.04953 .70032 L
.04159 .66026 L
.03551 .62019 L
.03121 .58013 L
.02865 .54006 L
.02779 .5 L
.02865 .45994 L
.03121 .41987 L
.03551 .37981 L
.04159 .33974 L
.04953 .29968 L
.05929 .2601 L
.05943 .25962 L
.07142 .21955 L
.0857 .17949 L
.09936 .14648 L
.10252 .13942 L
.12217 .09936 L
.13942 .06876 L
.14518 .05929 L
.17223 .01923 L
.98077 .01923 L
.98077 .98077 L
F
0 g
.17223 .98077 m
.14518 .94071 L
.13942 .93124 L
.12217 .90064 L
.10252 .86058 L
.09936 .85352 L
.0857 .82051 L
.07142 .78045 L
.05943 .74038 L
.05929 .7399 L
.04953 .70032 L
.04159 .66026 L
.03551 .62019 L
.03121 .58013 L
.02865 .54006 L
.02779 .5 L
.02865 .45994 L
.03121 .41987 L
.03551 .37981 L
.04159 .33974 L
.04953 .29968 L
.05929 .2601 L
.05943 .25962 L
.07142 .21955 L
.0857 .17949 L
.09936 .14648 L
.10252 .13942 L
.12217 .09936 L
.13942 .06876 L
.14518 .05929 L
.17223 .01923 L
s
.5 g
.22733 .98077 m
.21955 .97168 L
.19535 .94071 L
.17949 .91756 L
.16886 .90064 L
.14658 .86058 L
.13942 .8461 L
.12778 .82051 L
.11194 .78045 L
.09936 .74245 L
.09874 .74038 L
.08788 .70032 L
.0792 .66026 L
.07258 .62019 L
.06791 .58013 L
.06514 .54006 L
.06422 .5 L
.06514 .45994 L
.06791 .41987 L
.07258 .37981 L
.0792 .33974 L
.08788 .29968 L
.09874 .25962 L
.09936 .25755 L
.11194 .21955 L
.12778 .17949 L
.13942 .1539 L
.14658 .13942 L
.16886 .09936 L
.17949 .08244 L
.19535 .05929 L
.21955 .02832 L
.22733 .01923 L
.98077 .01923 L
.98077 .98077 L
F
0 g
.22733 .98077 m
.21955 .97168 L
.19535 .94071 L
.17949 .91756 L
.16886 .90064 L
.14658 .86058 L
.13942 .8461 L
.12778 .82051 L
.11194 .78045 L
.09936 .74245 L
.09874 .74038 L
.08788 .70032 L
.0792 .66026 L
.07258 .62019 L
.06791 .58013 L
.06514 .54006 L
.06422 .5 L
.06514 .45994 L
.06791 .41987 L
.07258 .37981 L
.0792 .33974 L
.08788 .29968 L
.09874 .25962 L
.09936 .25755 L
.11194 .21955 L
.12778 .17949 L
.13942 .1539 L
.14658 .13942 L
.16886 .09936 L
.17949 .08244 L
.19535 .05929 L
.21955 .02832 L
.22733 .01923 L
s
.6 g
.29691 .98077 m
.25962 .94502 L
.25565 .94071 L
.22331 .90064 L
.21955 .89537 L
.19698 .86058 L
.17949 .82889 L
.17527 .82051 L
.15722 .78045 L
.14234 .74038 L
.13942 .73154 L
.1302 .70032 L
.12056 .66026 L
.11322 .62019 L
.10807 .58013 L
.10502 .54006 L
.10401 .5 L
.10502 .45994 L
.10807 .41987 L
.11322 .37981 L
.12056 .33974 L
.1302 .29968 L
.13942 .26846 L
.14234 .25962 L
.15722 .21955 L
.17527 .17949 L
.17949 .17111 L
.19698 .13942 L
.21955 .10463 L
.22331 .09936 L
.25565 .05929 L
.25962 .05498 L
.29691 .01923 L
.98077 .01923 L
.98077 .98077 L
F
0 g
.29691 .98077 m
.25962 .94502 L
.25565 .94071 L
.22331 .90064 L
.21955 .89537 L
.19698 .86058 L
.17949 .82889 L
.17527 .82051 L
.15722 .78045 L
.14234 .74038 L
.13942 .73154 L
.1302 .70032 L
.12056 .66026 L
.11322 .62019 L
.10807 .58013 L
.10502 .54006 L
.10401 .5 L
.10502 .45994 L
.10807 .41987 L
.11322 .37981 L
.12056 .33974 L
.1302 .29968 L
.13942 .26846 L
.14234 .25962 L
.15722 .21955 L
.17527 .17949 L
.17949 .17111 L
.19698 .13942 L
.21955 .10463 L
.22331 .09936 L
.25565 .05929 L
.25962 .05498 L
.29691 .01923 L
s
.7 g
.40901 .98077 m
.37981 .96745 L
.33974 .94278 L
.33686 .94071 L
.29968 .90887 L
.29149 .90064 L
.25962 .86312 L
.2577 .86058 L
.23098 .82051 L
.21955 .80024 L
.20948 .78045 L
.19204 .74038 L
.17949 .70489 L
.17805 .70032 L
.16701 .66026 L
.15867 .62019 L
.15285 .58013 L
.1494 .54006 L
.14826 .5 L
.1494 .45994 L
.15285 .41987 L
.15867 .37981 L
.16701 .33974 L
.17805 .29968 L
.17949 .29511 L
.19204 .25962 L
.20948 .21955 L
.21955 .19976 L
.23098 .17949 L
.2577 .13942 L
.25962 .13688 L
.29149 .09936 L
.29968 .09113 L
.33686 .05929 L
.33974 .05722 L
.37981 .03255 L
.40901 .01923 L
.98077 .01923 L
.98077 .98077 L
F
0 g
.40901 .98077 m
.37981 .96745 L
.33974 .94278 L
.33686 .94071 L
.29968 .90887 L
.29149 .90064 L
.25962 .86312 L
.2577 .86058 L
.23098 .82051 L
.21955 .80024 L
.20948 .78045 L
.19204 .74038 L
.17949 .70489 L
.17805 .70032 L
.16701 .66026 L
.15867 .62019 L
.15285 .58013 L
.1494 .54006 L
.14826 .5 L
.1494 .45994 L
.15285 .41987 L
.15867 .37981 L
.16701 .33974 L
.17805 .29968 L
.17949 .29511 L
.19204 .25962 L
.20948 .21955 L
.21955 .19976 L
.23098 .17949 L
.2577 .13942 L
.25962 .13688 L
.29149 .09936 L
.29968 .09113 L
.33686 .05929 L
.33974 .05722 L
.37981 .03255 L
.40901 .01923 L
s
.8 g
.41987 .08963 m
.45994 .07806 L
.5 .07427 L
.54006 .07806 L
.58013 .08963 L
.60288 .09936 L
.62019 .10968 L
.66004 .13942 L
.66026 .13963 L
.69816 .17949 L
.70032 .18222 L
.72656 .21955 L
.74038 .2437 L
.74848 .25962 L
.76567 .29968 L
.77889 .33974 L
.78045 .34513 L
.7888 .37981 L
.79568 .41987 L
.79972 .45994 L
.80106 .5 L
.79972 .54006 L
.79568 .58013 L
.7888 .62019 L
.78045 .65487 L
.77889 .66026 L
.76567 .70032 L
.74848 .74038 L
.74038 .7563 L
.72656 .78045 L
.70032 .81778 L
.69816 .82051 L
.66026 .86037 L
.66004 .86058 L
.62019 .89032 L
.60288 .90064 L
.58013 .91037 L
.54006 .92194 L
.5 .92573 L
.45994 .92194 L
.41987 .91037 L
.39712 .90064 L
.37981 .89032 L
.33996 .86058 L
.33974 .86037 L
.30184 .82051 L
.29968 .81778 L
.27344 .78045 L
.25962 .7563 L
.25152 .74038 L
.23433 .70032 L
.22111 .66026 L
.21955 .65487 L
.2112 .62019 L
.20432 .58013 L
.20028 .54006 L
.19894 .5 L
.20028 .45994 L
.20432 .41987 L
.2112 .37981 L
.21955 .34513 L
.22111 .33974 L
.23433 .29968 L
.25152 .25962 L
.25962 .2437 L
.27344 .21955 L
.29968 .18222 L
.30184 .17949 L
.33974 .13963 L
.33996 .13942 L
.37981 .10968 L
.39712 .09936 L
F
0 g
.41987 .08963 m
.45994 .07806 L
.5 .07427 L
.54006 .07806 L
.58013 .08963 L
.60288 .09936 L
.62019 .10968 L
.66004 .13942 L
.66026 .13963 L
.69816 .17949 L
.70032 .18222 L
.72656 .21955 L
.74038 .2437 L
.74848 .25962 L
.76567 .29968 L
.77889 .33974 L
.78045 .34513 L
.7888 .37981 L
.79568 .41987 L
.79972 .45994 L
.80106 .5 L
.79972 .54006 L
.79568 .58013 L
.7888 .62019 L
.78045 .65487 L
.77889 .66026 L
.76567 .70032 L
.74848 .74038 L
.74038 .7563 L
.72656 .78045 L
.70032 .81778 L
.69816 .82051 L
.66026 .86037 L
.66004 .86058 L
.62019 .89032 L
.60288 .90064 L
.58013 .91037 L
.54006 .92194 L
.5 .92573 L
.45994 .92194 L
.41987 .91037 L
.39712 .90064 L
.37981 .89032 L
.33996 .86058 L
.33974 .86037 L
.30184 .82051 L
.29968 .81778 L
.27344 .78045 L
.25962 .7563 L
.25152 .74038 L
Mistroke
.23433 .70032 L
.22111 .66026 L
.21955 .65487 L
.2112 .62019 L
.20432 .58013 L
.20028 .54006 L
.19894 .5 L
.20028 .45994 L
.20432 .41987 L
.2112 .37981 L
.21955 .34513 L
.22111 .33974 L
.23433 .29968 L
.25152 .25962 L
.25962 .2437 L
.27344 .21955 L
.29968 .18222 L
.30184 .17949 L
.33974 .13963 L
.33996 .13942 L
.37981 .10968 L
.39712 .09936 L
.41987 .08963 L
Mfstroke
.9 g
.45994 .16556 m
.5 .1608 L
.54006 .16556 L
.57847 .17949 L
.58013 .18031 L
.62019 .20645 L
.63522 .21955 L
.66026 .24761 L
.66931 .25962 L
.69363 .29968 L
.70032 .31334 L
.71145 .33974 L
.72436 .37981 L
.7331 .41987 L
.73818 .45994 L
.73984 .5 L
.73818 .54006 L
.7331 .58013 L
.72436 .62019 L
.71145 .66026 L
.70032 .68666 L
.69363 .70032 L
.66931 .74038 L
.66026 .75239 L
.63522 .78045 L
.62019 .79355 L
.58013 .81969 L
.57847 .82051 L
.54006 .83444 L
.5 .8392 L
.45994 .83444 L
.42153 .82051 L
.41987 .81969 L
.37981 .79355 L
.36478 .78045 L
.33974 .75239 L
.33069 .74038 L
.30637 .70032 L
.29968 .68666 L
.28855 .66026 L
.27564 .62019 L
.2669 .58013 L
.26182 .54006 L
.26016 .5 L
.26182 .45994 L
.2669 .41987 L
.27564 .37981 L
.28855 .33974 L
.29968 .31334 L
.30637 .29968 L
.33069 .25962 L
.33974 .24761 L
.36478 .21955 L
.37981 .20645 L
.41987 .18031 L
.42153 .17949 L
F
0 g
.45994 .16556 m
.5 .1608 L
.54006 .16556 L
.57847 .17949 L
.58013 .18031 L
.62019 .20645 L
.63522 .21955 L
.66026 .24761 L
.66931 .25962 L
.69363 .29968 L
.70032 .31334 L
.71145 .33974 L
.72436 .37981 L
.7331 .41987 L
.73818 .45994 L
.73984 .5 L
.73818 .54006 L
.7331 .58013 L
.72436 .62019 L
.71145 .66026 L
.70032 .68666 L
.69363 .70032 L
.66931 .74038 L
.66026 .75239 L
.63522 .78045 L
.62019 .79355 L
.58013 .81969 L
.57847 .82051 L
.54006 .83444 L
.5 .8392 L
.45994 .83444 L
.42153 .82051 L
.41987 .81969 L
.37981 .79355 L
.36478 .78045 L
.33974 .75239 L
.33069 .74038 L
.30637 .70032 L
.29968 .68666 L
.28855 .66026 L
.27564 .62019 L
.2669 .58013 L
.26182 .54006 L
.26016 .5 L
.26182 .45994 L
.2669 .41987 L
.27564 .37981 L
.28855 .33974 L
.29968 .31334 L
.30637 .29968 L
Mistroke
.33069 .25962 L
.33974 .24761 L
.36478 .21955 L
.37981 .20645 L
.41987 .18031 L
.42153 .17949 L
.45994 .16556 L
Mfstroke
1 g
.45994 .28628 m
.5 .27888 L
.54006 .28628 L
.56605 .29968 L
.58013 .3101 L
.60861 .33974 L
.62019 .35835 L
.63143 .37981 L
.646 .41987 L
.6539 .45994 L
.65642 .5 L
.6539 .54006 L
.646 .58013 L
.63143 .62019 L
.62019 .64165 L
.60861 .66026 L
.58013 .6899 L
.56605 .70032 L
.54006 .71372 L
.5 .72112 L
.45994 .71372 L
.43395 .70032 L
.41987 .6899 L
.39139 .66026 L
.37981 .64165 L
.36857 .62019 L
.354 .58013 L
.3461 .54006 L
.34358 .5 L
.3461 .45994 L
.354 .41987 L
.36857 .37981 L
.37981 .35835 L
.39139 .33974 L
.41987 .3101 L
.43395 .29968 L
F
0 g
.45994 .28628 m
.5 .27888 L
.54006 .28628 L
.56605 .29968 L
.58013 .3101 L
.60861 .33974 L
.62019 .35835 L
.63143 .37981 L
.646 .41987 L
.6539 .45994 L
.65642 .5 L
.6539 .54006 L
.646 .58013 L
.63143 .62019 L
.62019 .64165 L
.60861 .66026 L
.58013 .6899 L
.56605 .70032 L
.54006 .71372 L
.5 .72112 L
.45994 .71372 L
.43395 .70032 L
.41987 .6899 L
.39139 .66026 L
.37981 .64165 L
.36857 .62019 L
.354 .58013 L
.3461 .54006 L
.34358 .5 L
.3461 .45994 L
.354 .41987 L
.36857 .37981 L
.37981 .35835 L
.39139 .33974 L
.41987 .3101 L
.43395 .29968 L
.45994 .28628 L
s
.6 g
.59099 .98077 m
.62019 .96745 L
.66026 .94278 L
.66314 .94071 L
.70032 .90887 L
.70851 .90064 L
.74038 .86312 L
.7423 .86058 L
.76902 .82051 L
.78045 .80024 L
.79052 .78045 L
.80796 .74038 L
.82051 .70489 L
.82195 .70032 L
.83299 .66026 L
.84133 .62019 L
.84715 .58013 L
.8506 .54006 L
.85174 .5 L
.8506 .45994 L
.84715 .41987 L
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.99375 .74038 m
1 .74038 L
s
.99375 .98077 m
1 .98077 L
s
.125 Mabswid
.99625 .06731 m
1 .06731 L
s
.99625 .11538 m
1 .11538 L
s
.99625 .16346 m
1 .16346 L
s
.99625 .21154 m
1 .21154 L
s
.99625 .30769 m
1 .30769 L
s
.99625 .35577 m
1 .35577 L
s
.99625 .40385 m
1 .40385 L
s
.99625 .45192 m
1 .45192 L
s
.99625 .54808 m
1 .54808 L
s
.99625 .59615 m
1 .59615 L
s
.99625 .64423 m
1 .64423 L
s
.99625 .69231 m
1 .69231 L
s
.99625 .78846 m
1 .78846 L
s
.99625 .83654 m
1 .83654 L
s
.99625 .88462 m
1 .88462 L
s
.99625 .93269 m
1 .93269 L
s
.25 Mabswid
1 0 m
1 1 L
s
0 0 m
1 0 L
1 1 L
0 1 L
closepath
clip
newpath
.8 g
.01923 .98077 m
.98077 .98077 L
.98077 .01923 L
.01923 .01923 L
F
0 g
.5 Mabswid
.7 g
.01923 .22871 m
.02989 .21955 L
.05929 .19659 L
.08423 .17949 L
.09936 .16999 L
.13942 .14765 L
.15606 .13942 L
.17949 .12879 L
.21955 .11291 L
.25962 .09967 L
.26064 .09936 L
.29968 .08879 L
.33974 .0801 L
.37981 .07346 L
.41987 .06878 L
.45994 .066 L
.5 .06508 L
.54006 .066 L
.58013 .06878 L
.62019 .07346 L
.66026 .0801 L
.70032 .08879 L
.73936 .09936 L
.74038 .09967 L
.78045 .11291 L
.82051 .12879 L
.84394 .13942 L
.86058 .14765 L
.90064 .16999 L
.91577 .17949 L
.94071 .19659 L
.97011 .21955 L
.98077 .22871 L
.98077 .98077 L
.01923 .98077 L
F
0 g
.01923 .22871 m
.02989 .21955 L
.05929 .19659 L
.08423 .17949 L
.09936 .16999 L
.13942 .14765 L
.15606 .13942 L
.17949 .12879 L
.21955 .11291 L
.25962 .09967 L
.26064 .09936 L
.29968 .08879 L
.33974 .0801 L
.37981 .07346 L
.41987 .06878 L
.45994 .066 L
.5 .06508 L
.54006 .066 L
.58013 .06878 L
.62019 .07346 L
.66026 .0801 L
.70032 .08879 L
.73936 .09936 L
.74038 .09967 L
.78045 .11291 L
.82051 .12879 L
.84394 .13942 L
.86058 .14765 L
.90064 .16999 L
.91577 .17949 L
.94071 .19659 L
.97011 .21955 L
.98077 .22871 L
s
.6 g
.01923 .33303 m
.04547 .29968 L
.05929 .28474 L
.08613 .25962 L
.09936 .24864 L
.13942 .21996 L
.14007 .21955 L
.17949 .19657 L
.21469 .17949 L
.21955 .17735 L
.25962 .16157 L
.29968 .14877 L
.33618 .13942 L
.33974 .13863 L
.37981 .13094 L
.41987 .12554 L
.45994 .12234 L
.5 .12128 L
.54006 .12234 L
.58013 .12554 L
.62019 .13094 L
.66026 .13863 L
.66382 .13942 L
.70032 .14877 L
.74038 .16157 L
.78045 .17735 L
.78531 .17949 L
.82051 .19657 L
.85993 .21955 L
.86058 .21996 L
.90064 .24864 L
.91387 .25962 L
.94071 .28474 L
.95453 .29968 L
.98077 .33303 L
.98077 .98077 L
.01923 .98077 L
F
0 g
.01923 .33303 m
.04547 .29968 L
.05929 .28474 L
.08613 .25962 L
.09936 .24864 L
.13942 .21996 L
.14007 .21955 L
.17949 .19657 L
.21469 .17949 L
.21955 .17735 L
.25962 .16157 L
.29968 .14877 L
.33618 .13942 L
.33974 .13863 L
.37981 .13094 L
.41987 .12554 L
.45994 .12234 L
.5 .12128 L
.54006 .12234 L
.58013 .12554 L
.62019 .13094 L
.66026 .13863 L
.66382 .13942 L
.70032 .14877 L
.74038 .16157 L
.78045 .17735 L
.78531 .17949 L
.82051 .19657 L
.85993 .21955 L
.86058 .21996 L
.90064 .24864 L
.91387 .25962 L
.94071 .28474 L
.95453 .29968 L
.98077 .33303 L
s
.7 g
.01923 .66697 m
.04547 .70032 L
.05929 .71526 L
.08613 .74038 L
.09936 .75136 L
.13942 .78004 L
.14007 .78045 L
.17949 .80343 L
.21469 .82051 L
.21955 .82265 L
.25962 .83843 L
.29968 .85123 L
.33618 .86058 L
.33974 .86137 L
.37981 .86906 L
.41987 .87446 L
.45994 .87766 L
.5 .87872 L
.54006 .87766 L
.58013 .87446 L
.62019 .86906 L
.66026 .86137 L
.66382 .86058 L
.70032 .85123 L
.74038 .83843 L
.78045 .82265 L
.78531 .82051 L
.82051 .80343 L
.85993 .78045 L
.86058 .78004 L
.90064 .75136 L
.91387 .74038 L
.94071 .71526 L
.95453 .70032 L
.98077 .66697 L
.98077 .98077 L
.01923 .98077 L
F
0 g
.01923 .66697 m
.04547 .70032 L
.05929 .71526 L
.08613 .74038 L
.09936 .75136 L
.13942 .78004 L
.14007 .78045 L
.17949 .80343 L
.21469 .82051 L
.21955 .82265 L
.25962 .83843 L
.29968 .85123 L
.33618 .86058 L
.33974 .86137 L
.37981 .86906 L
.41987 .87446 L
.45994 .87766 L
.5 .87872 L
.54006 .87766 L
.58013 .87446 L
.62019 .86906 L
.66026 .86137 L
.66382 .86058 L
.70032 .85123 L
.74038 .83843 L
.78045 .82265 L
.78531 .82051 L
.82051 .80343 L
.85993 .78045 L
.86058 .78004 L
.90064 .75136 L
.91387 .74038 L
.94071 .71526 L
.95453 .70032 L
.98077 .66697 L
s
.8 g
.01923 .77129 m
.02989 .78045 L
.05929 .80341 L
.08423 .82051 L
.09936 .83001 L
.13942 .85235 L
.15606 .86058 L
.17949 .87121 L
.21955 .88709 L
.25962 .90033 L
.26064 .90064 L
.29968 .91121 L
.33974 .9199 L
.37981 .92654 L
.41987 .93122 L
.45994 .934 L
.5 .93492 L
.54006 .934 L
.58013 .93122 L
.62019 .92654 L
.66026 .9199 L
.70032 .91121 L
.73936 .90064 L
.74038 .90033 L
.78045 .88709 L
.82051 .87121 L
.84394 .86058 L
.86058 .85235 L
.90064 .83001 L
.91577 .82051 L
.94071 .80341 L
.97011 .78045 L
.98077 .77129 L
.98077 .98077 L
.01923 .98077 L
F
0 g
.01923 .77129 m
.02989 .78045 L
.05929 .80341 L
.08423 .82051 L
.09936 .83001 L
.13942 .85235 L
.15606 .86058 L
.17949 .87121 L
.21955 .88709 L
.25962 .90033 L
.26064 .90064 L
.29968 .91121 L
.33974 .9199 L
.37981 .92654 L
.41987 .93122 L
.45994 .934 L
.5 .93492 L
.54006 .934 L
.58013 .93122 L
.62019 .92654 L
.66026 .9199 L
.70032 .91121 L
.73936 .90064 L
.74038 .90033 L
.78045 .88709 L
.82051 .87121 L
.84394 .86058 L
.86058 .85235 L
.90064 .83001 L
.91577 .82051 L
.94071 .80341 L
.97011 .78045 L
.98077 .77129 L
s
.9 g
.01923 .14554 m
.02843 .13942 L
.05929 .12039 L
.09827 .09936 L
.09936 .09881 L
.13942 .08024 L
.17949 .06429 L
.19342 .05929 L
.21955 .05069 L
.25962 .03923 L
.29968 .02974 L
.33974 .02212 L
.35775 .01923 L
.01923 .01923 L
F
0 g
.01923 .14554 m
.02843 .13942 L
.05929 .12039 L
.09827 .09936 L
.09936 .09881 L
.13942 .08024 L
.17949 .06429 L
.19342 .05929 L
.21955 .05069 L
.25962 .03923 L
.29968 .02974 L
.33974 .02212 L
.35775 .01923 L
s
.9 g
.01923 .85446 m
.02843 .86058 L
.05929 .87961 L
.09827 .90064 L
.09936 .90119 L
.13942 .91976 L
.17949 .93571 L
.19342 .94071 L
.21955 .94931 L
.25962 .96077 L
.29968 .97026 L
.33974 .97788 L
.35775 .98077 L
.01923 .98077 L
F
0 g
.01923 .85446 m
.02843 .86058 L
.05929 .87961 L
.09827 .90064 L
.09936 .90119 L
.13942 .91976 L
.17949 .93571 L
.19342 .94071 L
.21955 .94931 L
.25962 .96077 L
.29968 .97026 L
.33974 .97788 L
.35775 .98077 L
s
1 g
.01923 .07104 m
.04098 .05929 L
.05929 .05004 L
.09936 .03169 L
.13003 .01923 L
.01923 .01923 L
F
0 g
.01923 .07104 m
.04098 .05929 L
.05929 .05004 L
.09936 .03169 L
.13003 .01923 L
s
1 g
.01923 .92896 m
.04098 .94071 L
.05929 .94996 L
.09936 .96831 L
.13003 .98077 L
.01923 .98077 L
F
0 g
.01923 .92896 m
.04098 .94071 L
.05929 .94996 L
.09936 .96831 L
.13003 .98077 L
s
.5 g
.45994 .17873 m
.5 .17749 L
.54006 .17873 L
.5507 .17949 L
.58013 .1825 L
.62019 .18889 L
.66026 .19805 L
.70032 .21025 L
.72528 .21955 L
.74038 .22591 L
.78045 .24564 L
.80409 .25962 L
.82051 .27051 L
.85745 .29968 L
.86058 .30247 L
.89581 .33974 L
.90064 .34572 L
.92325 .37981 L
.94071 .41668 L
.9418 .41987 L
.95257 .45994 L
.9561 .5 L
.95257 .54006 L
.9418 .58013 L
.94071 .58332 L
.92325 .62019 L
.90064 .65428 L
.89581 .66026 L
.86058 .69753 L
.85745 .70032 L
.82051 .72949 L
.80409 .74038 L
.78045 .75436 L
.74038 .77409 L
.72528 .78045 L
.70032 .78975 L
.66026 .80195 L
.62019 .81111 L
.58013 .8175 L
.5507 .82051 L
.54006 .82127 L
.5 .82251 L
.45994 .82127 L
.4493 .82051 L
.41987 .8175 L
.37981 .81111 L
.33974 .80195 L
.29968 .78975 L
.27472 .78045 L
.25962 .77409 L
.21955 .75436 L
.19591 .74038 L
.17949 .72949 L
.14255 .70032 L
.13942 .69753 L
.10419 .66026 L
.09936 .65428 L
.07675 .62019 L
.05929 .58332 L
.0582 .58013 L
.04743 .54006 L
.0439 .5 L
.04743 .45994 L
.0582 .41987 L
.05929 .41668 L
.07675 .37981 L
.09936 .34572 L
.10419 .33974 L
.13942 .30247 L
.14255 .29968 L
.17949 .27051 L
.19591 .25962 L
.21955 .24564 L
.25962 .22591 L
.27472 .21955 L
.29968 .21025 L
.33974 .19805 L
.37981 .18889 L
.41987 .1825 L
.4493 .17949 L
F
0 g
.45994 .17873 m
.5 .17749 L
.54006 .17873 L
.5507 .17949 L
.58013 .1825 L
.62019 .18889 L
.66026 .19805 L
.70032 .21025 L
.72528 .21955 L
.74038 .22591 L
.78045 .24564 L
.80409 .25962 L
.82051 .27051 L
.85745 .29968 L
.86058 .30247 L
.89581 .33974 L
.90064 .34572 L
.92325 .37981 L
.94071 .41668 L
.9418 .41987 L
.95257 .45994 L
.9561 .5 L
.95257 .54006 L
.9418 .58013 L
.94071 .58332 L
.92325 .62019 L
.90064 .65428 L
.89581 .66026 L
.86058 .69753 L
.85745 .70032 L
.82051 .72949 L
.80409 .74038 L
.78045 .75436 L
.74038 .77409 L
.72528 .78045 L
.70032 .78975 L
.66026 .80195 L
.62019 .81111 L
.58013 .8175 L
.5507 .82051 L
.54006 .82127 L
.5 .82251 L
.45994 .82127 L
.4493 .82051 L
.41987 .8175 L
.37981 .81111 L
.33974 .80195 L
.29968 .78975 L
.27472 .78045 L
.25962 .77409 L
Mistroke
.21955 .75436 L
.19591 .74038 L
.17949 .72949 L
.14255 .70032 L
.13942 .69753 L
.10419 .66026 L
.09936 .65428 L
.07675 .62019 L
.05929 .58332 L
.0582 .58013 L
.04743 .54006 L
.0439 .5 L
.04743 .45994 L
.0582 .41987 L
.05929 .41668 L
.07675 .37981 L
.09936 .34572 L
.10419 .33974 L
.13942 .30247 L
.14255 .29968 L
.17949 .27051 L
.19591 .25962 L
.21955 .24564 L
.25962 .22591 L
.27472 .21955 L
.29968 .21025 L
.33974 .19805 L
.37981 .18889 L
.41987 .1825 L
.4493 .17949 L
.45994 .17873 L
Mfstroke
.4 g
.33974 .259 m
.37981 .24762 L
.41987 .23979 L
.45994 .2352 L
.5 .23369 L
.54006 .2352 L
.58013 .23979 L
.62019 .24762 L
.66026 .259 L
.6621 .25962 L
.70032 .27447 L
.74038 .29498 L
.74817 .29968 L
.78045 .32217 L
.8008 .33974 L
.82051 .35991 L
.83608 .37981 L
.85916 .41987 L
.86058 .42071 L
.87233 .45994 L
.87662 .5 L
.87233 .54006 L
.86058 .57929 L
.85916 .58013 L
.83608 .62019 L
.82051 .64009 L
.8008 .66026 L
.78045 .67783 L
.74817 .70032 L
.74038 .70502 L
.70032 .72553 L
.6621 .74038 L
.66026 .741 L
.62019 .75238 L
.58013 .76021 L
.54006 .7648 L
.5 .76631 L
.45994 .7648 L
.41987 .76021 L
.37981 .75238 L
.33974 .741 L
.3379 .74038 L
.29968 .72553 L
.25962 .70502 L
.25183 .70032 L
.21955 .67783 L
.1992 .66026 L
.17949 .64009 L
.16392 .62019 L
.14084 .58013 L
.13942 .57929 L
.12767 .54006 L
.12338 .5 L
.12767 .45994 L
.13942 .42071 L
.14084 .41987 L
.16392 .37981 L
.17949 .35991 L
.1992 .33974 L
.21955 .32217 L
.25183 .29968 L
.25962 .29498 L
.29968 .27447 L
.3379 .25962 L
F
0 g
.33974 .259 m
.37981 .24762 L
.41987 .23979 L
.45994 .2352 L
.5 .23369 L
.54006 .2352 L
.58013 .23979 L
.62019 .24762 L
.66026 .259 L
.6621 .25962 L
.70032 .27447 L
.74038 .29498 L
.74817 .29968 L
.78045 .32217 L
.8008 .33974 L
.82051 .35991 L
.83608 .37981 L
.85916 .41987 L
.86058 .42071 L
.87233 .45994 L
.87662 .5 L
.87233 .54006 L
.86058 .57929 L
.85916 .58013 L
.83608 .62019 L
.82051 .64009 L
.8008 .66026 L
.78045 .67783 L
.74817 .70032 L
.74038 .70502 L
.70032 .72553 L
.6621 .74038 L
.66026 .741 L
.62019 .75238 L
.58013 .76021 L
.54006 .7648 L
.5 .76631 L
.45994 .7648 L
.41987 .76021 L
.37981 .75238 L
.33974 .741 L
.3379 .74038 L
.29968 .72553 L
.25962 .70502 L
.25183 .70032 L
.21955 .67783 L
.1992 .66026 L
.17949 .64009 L
.16392 .62019 L
.14084 .58013 L
Mistroke
.13942 .57929 L
.12767 .54006 L
.12338 .5 L
.12767 .45994 L
.13942 .42071 L
.14084 .41987 L
.16392 .37981 L
.17949 .35991 L
.1992 .33974 L
.21955 .32217 L
.25183 .29968 L
.25962 .29498 L
.29968 .27447 L
.3379 .25962 L
.33974 .259 L
Mfstroke
.3 g
.41987 .29768 m
.45994 .29182 L
.5 .2899 L
.54006 .29182 L
.58013 .29768 L
.59025 .29968 L
.62019 .30784 L
.66026 .32304 L
.69221 .33974 L
.70032 .34476 L
.74038 .37645 L
.74371 .37981 L
.77467 .41987 L
.78045 .42452 L
.79168 .45994 L
.79713 .5 L
.79168 .54006 L
.78045 .57548 L
.77467 .58013 L
.74371 .62019 L
.74038 .62355 L
.70032 .65524 L
.69221 .66026 L
.66026 .67696 L
.62019 .69216 L
.59025 .70032 L
.58013 .70232 L
.54006 .70818 L
.5 .7101 L
.45994 .70818 L
.41987 .70232 L
.40975 .70032 L
.37981 .69216 L
.33974 .67696 L
.30779 .66026 L
.29968 .65524 L
.25962 .62355 L
.25629 .62019 L
.22533 .58013 L
.21955 .57548 L
.20832 .54006 L
.20287 .5 L
.20832 .45994 L
.21955 .42452 L
.22533 .41987 L
.25629 .37981 L
.25962 .37645 L
.29968 .34476 L
.30779 .33974 L
.33974 .32304 L
.37981 .30784 L
.40975 .29968 L
F
0 g
.41987 .29768 m
.45994 .29182 L
.5 .2899 L
.54006 .29182 L
.58013 .29768 L
.59025 .29968 L
.62019 .30784 L
.66026 .32304 L
.69221 .33974 L
.70032 .34476 L
.74038 .37645 L
.74371 .37981 L
.77467 .41987 L
.78045 .42452 L
.79168 .45994 L
.79713 .5 L
.79168 .54006 L
.78045 .57548 L
.77467 .58013 L
.74371 .62019 L
.74038 .62355 L
.70032 .65524 L
.69221 .66026 L
.66026 .67696 L
.62019 .69216 L
.59025 .70032 L
.58013 .70232 L
.54006 .70818 L
.5 .7101 L
.45994 .70818 L
.41987 .70232 L
.40975 .70032 L
.37981 .69216 L
.33974 .67696 L
.30779 .66026 L
.29968 .65524 L
.25962 .62355 L
.25629 .62019 L
.22533 .58013 L
.21955 .57548 L
.20832 .54006 L
.20287 .5 L
.20832 .45994 L
.21955 .42452 L
.22533 .41987 L
.25629 .37981 L
.25962 .37645 L
.29968 .34476 L
.30779 .33974 L
.33974 .32304 L
Mistroke
.37981 .30784 L
.40975 .29968 L
.41987 .29768 L
Mfstroke
.2 g
.37981 .37164 m
.41987 .35687 L
.45994 .34873 L
.5 .3461 L
.54006 .34873 L
.58013 .35687 L
.62019 .37164 L
.63612 .37981 L
.66026 .3953 L
.68585 .41987 L
.70032 .43294 L
.71014 .45994 L
.71764 .5 L
.71014 .54006 L
.70032 .56706 L
.68585 .58013 L
.66026 .6047 L
.63612 .62019 L
.62019 .62836 L
.58013 .64313 L
.54006 .65127 L
.5 .6539 L
.45994 .65127 L
.41987 .64313 L
.37981 .62836 L
.36388 .62019 L
.33974 .6047 L
.31415 .58013 L
.29968 .56706 L
.28986 .54006 L
.28236 .5 L
.28986 .45994 L
.29968 .43294 L
.31415 .41987 L
.33974 .3953 L
.36388 .37981 L
F
0 g
.37981 .37164 m
.41987 .35687 L
.45994 .34873 L
.5 .3461 L
.54006 .34873 L
.58013 .35687 L
.62019 .37164 L
.63612 .37981 L
.66026 .3953 L
.68585 .41987 L
.70032 .43294 L
.71014 .45994 L
.71764 .5 L
.71014 .54006 L
.70032 .56706 L
.68585 .58013 L
.66026 .6047 L
.63612 .62019 L
.62019 .62836 L
.58013 .64313 L
.54006 .65127 L
.5 .6539 L
.45994 .65127 L
.41987 .64313 L
.37981 .62836 L
.36388 .62019 L
.33974 .6047 L
.31415 .58013 L
.29968 .56706 L
.28986 .54006 L
.28236 .5 L
.28986 .45994 L
.29968 .43294 L
.31415 .41987 L
.33974 .3953 L
.36388 .37981 L
.37981 .37164 L
s
.1 g
.45994 .40643 m
.5 .40231 L
.54006 .40643 L
.5797 .41987 L
.58013 .42022 L
.62019 .44843 L
.62603 .45994 L
.63816 .5 L
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future labs, and therefore know what we'd like students to learn about \
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scratch. I'm not entirely happy with them, but they do provide some insight \
with the cross sections. Please send me any comments or questions!\n\n\
Update: I rewrote parts of this lab in January 2004. I edited exercises, \
removed references to the unsupported and undocumented RealTime3D package, \
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quadric surfaces. The old version was a regurgitation of the textbook. The \
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specifically required by the license, I'd appreciate it if you let me know if \
you use parts of our labs, just so I can keep track of it. Please send me \
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